metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D244, C4⋊D61, C61⋊1D4, C244⋊1C2, D122⋊1C2, C2.4D122, C122.3C22, sometimes denoted D488 or Dih244 or Dih488, SmallGroup(488,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D244
G = < a,b | a244=b2=1, bab=a-1 >
Smallest permutation representation of D244
►On 244 points
Generators in S244
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244)
(1 183)(2 182)(3 181)(4 180)(5 179)(6 178)(7 177)(8 176)(9 175)(10 174)(11 173)(12 172)(13 171)(14 170)(15 169)(16 168)(17 167)(18 166)(19 165)(20 164)(21 163)(22 162)(23 161)(24 160)(25 159)(26 158)(27 157)(28 156)(29 155)(30 154)(31 153)(32 152)(33 151)(34 150)(35 149)(36 148)(37 147)(38 146)(39 145)(40 144)(41 143)(42 142)(43 141)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(51 133)(52 132)(53 131)(54 130)(55 129)(56 128)(57 127)(58 126)(59 125)(60 124)(61 123)(62 122)(63 121)(64 120)(65 119)(66 118)(67 117)(68 116)(69 115)(70 114)(71 113)(72 112)(73 111)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(184 244)(185 243)(186 242)(187 241)(188 240)(189 239)(190 238)(191 237)(192 236)(193 235)(194 234)(195 233)(196 232)(197 231)(198 230)(199 229)(200 228)(201 227)(202 226)(203 225)(204 224)(205 223)(206 222)(207 221)(208 220)(209 219)(210 218)(211 217)(212 216)(213 215)
G:=sub<Sym(244)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244), (1,183)(2,182)(3,181)(4,180)(5,179)(6,178)(7,177)(8,176)(9,175)(10,174)(11,173)(12,172)(13,171)(14,170)(15,169)(16,168)(17,167)(18,166)(19,165)(20,164)(21,163)(22,162)(23,161)(24,160)(25,159)(26,158)(27,157)(28,156)(29,155)(30,154)(31,153)(32,152)(33,151)(34,150)(35,149)(36,148)(37,147)(38,146)(39,145)(40,144)(41,143)(42,142)(43,141)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,133)(52,132)(53,131)(54,130)(55,129)(56,128)(57,127)(58,126)(59,125)(60,124)(61,123)(62,122)(63,121)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(184,244)(185,243)(186,242)(187,241)(188,240)(189,239)(190,238)(191,237)(192,236)(193,235)(194,234)(195,233)(196,232)(197,231)(198,230)(199,229)(200,228)(201,227)(202,226)(203,225)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)(211,217)(212,216)(213,215)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244), (1,183)(2,182)(3,181)(4,180)(5,179)(6,178)(7,177)(8,176)(9,175)(10,174)(11,173)(12,172)(13,171)(14,170)(15,169)(16,168)(17,167)(18,166)(19,165)(20,164)(21,163)(22,162)(23,161)(24,160)(25,159)(26,158)(27,157)(28,156)(29,155)(30,154)(31,153)(32,152)(33,151)(34,150)(35,149)(36,148)(37,147)(38,146)(39,145)(40,144)(41,143)(42,142)(43,141)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,133)(52,132)(53,131)(54,130)(55,129)(56,128)(57,127)(58,126)(59,125)(60,124)(61,123)(62,122)(63,121)(64,120)(65,119)(66,118)(67,117)(68,116)(69,115)(70,114)(71,113)(72,112)(73,111)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,103)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(184,244)(185,243)(186,242)(187,241)(188,240)(189,239)(190,238)(191,237)(192,236)(193,235)(194,234)(195,233)(196,232)(197,231)(198,230)(199,229)(200,228)(201,227)(202,226)(203,225)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)(211,217)(212,216)(213,215) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244)], [(1,183),(2,182),(3,181),(4,180),(5,179),(6,178),(7,177),(8,176),(9,175),(10,174),(11,173),(12,172),(13,171),(14,170),(15,169),(16,168),(17,167),(18,166),(19,165),(20,164),(21,163),(22,162),(23,161),(24,160),(25,159),(26,158),(27,157),(28,156),(29,155),(30,154),(31,153),(32,152),(33,151),(34,150),(35,149),(36,148),(37,147),(38,146),(39,145),(40,144),(41,143),(42,142),(43,141),(44,140),(45,139),(46,138),(47,137),(48,136),(49,135),(50,134),(51,133),(52,132),(53,131),(54,130),(55,129),(56,128),(57,127),(58,126),(59,125),(60,124),(61,123),(62,122),(63,121),(64,120),(65,119),(66,118),(67,117),(68,116),(69,115),(70,114),(71,113),(72,112),(73,111),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,103),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(184,244),(185,243),(186,242),(187,241),(188,240),(189,239),(190,238),(191,237),(192,236),(193,235),(194,234),(195,233),(196,232),(197,231),(198,230),(199,229),(200,228),(201,227),(202,226),(203,225),(204,224),(205,223),(206,222),(207,221),(208,220),(209,219),(210,218),(211,217),(212,216),(213,215)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 61A | ··· | 61AD | 122A | ··· | 122AD | 244A | ··· | 244BH |
| order | 1 | 2 | 2 | 2 | 4 | 61 | ··· | 61 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 122 | 122 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D61 | D122 | D244 |
| kernel | D244 | C244 | D122 | C61 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 30 | 30 | 60 |
Matrix representation of D244 ►in GL2(𝔽733) generated by
| 313 | 605 |
| 270 | 461 |
| 506 | 112 |
| 430 | 227 |
G:=sub<GL(2,GF(733))| [313,270,605,461],[506,430,112,227] >;
D244 in GAP, Magma, Sage, TeX
D_{244} % in TeX
G:=Group("D244"); // GroupNames label
G:=SmallGroup(488,6);
// by ID
G=gap.SmallGroup(488,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,49,21,7683]);
// Polycyclic
G:=Group<a,b|a^244=b^2=1,b*a*b=a^-1>;
// generators/relations
Export